Finite Group Databasebeta

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C₂×#1370 #12003

order	256
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₈	D₄
commutator	C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₈	C₂³
socle, prime radical	C₂³	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2);(4,7)(5,8)(9,10)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(3,4,8,10,6,9,5,7)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#92911, #92920, #92923, …
maximal normal subgroup of	#92911, #92920, #92923, …
maximal quotient of	C₂×#7100, C₂×#7101, #122930, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4991);